Sr Examen
Integral de sqrt(a^2-x^2)/x dx
Solución
Ha introducido
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a / | | _________ | / 2 2 | \/ a - x | ------------ dx | x | / 0
$$\int\limits_{0}^{a} \frac{\sqrt{a^{2} - x^{2}}}{x}\, dx$$
Integral(sqrt(a^2 - x^2)/x, (x, 0, a))
Respuesta (Indefinida)
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// 2 | 2| \
|| /a\ x a |a | |
||- a*acosh|-| - --------------- + ----------------- for |--| > 1|
|| \x/ _________ _________ | 2| |
/ || / 2 / 2 |x | |
| || / a / a |
| _________ || / -1 + -- x* / -1 + -- |
| / 2 2 || / 2 / 2 |
| \/ a - x || \/ x \/ x |
| ------------ dx = C + |< |
| x || 2 |
| || /a\ I*x I*a |
/ || I*a*asin|-| + -------------- - ---------------- otherwise |
|| \x/ ________ ________ |
|| / 2 / 2 |
|| / a / a |
|| / 1 - -- x* / 1 - -- |
|| / 2 / 2 |
\\ \/ x \/ x /
$$\int \frac{\sqrt{a^{2} - x^{2}}}{x}\, dx = C + \begin{cases} \frac{a^{2}}{x \sqrt{\frac{a^{2}}{x^{2}} - 1}} - a \operatorname{acosh}{\left(\frac{a}{x} \right)} - \frac{x}{\sqrt{\frac{a^{2}}{x^{2}} - 1}} & \text{for}\: \left|{\frac{a^{2}}{x^{2}}}\right| > 1 \\- \frac{i a^{2}}{x \sqrt{- \frac{a^{2}}{x^{2}} + 1}} + i a \operatorname{asin}{\left(\frac{a}{x} \right)} + \frac{i x}{\sqrt{- \frac{a^{2}}{x^{2}} + 1}} & \text{otherwise} \end{cases}$$
Respuesta
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/ 0 | / | | | | / 4 2 | | | 1 a a 2 |1 | | | |- --------------- + --------------- - --------------- for a *|--| > 1 | | | _________ 3/2 3/2 | 2| | | | / 2 / 2\ / 2\ |x | | | | / a 4 | a | 2 | a | | | | / -1 + -- x *|-1 + --| x *|-1 + --| | | | / 2 | 2| | 2| | | | \/ x \ x / \ x / |- | < dx for a < 0 | | | 4 2 | | | I I*a I*a | | | -------------- + -------------- - -------------- otherwise | | | ________ 3/2 3/2 | | | / 2 / 2\ / 2\ | | | / a 4 | a | 2 | a | | | | / 1 - -- x *|1 - --| x *|1 - --| | | | / 2 | 2| | 2| | | \ \/ x \ x / \ x / | | | / | a < | a | / | | | | / 4 2 | | | 1 a a 2 |1 | | | |- --------------- + --------------- - --------------- for a *|--| > 1 | | | _________ 3/2 3/2 | 2| | | | / 2 / 2\ / 2\ |x | | | | / a 4 | a | 2 | a | | | | / -1 + -- x *|-1 + --| x *|-1 + --| | | | / 2 | 2| | 2| | | | \/ x \ x / \ x / | | < dx otherwise | | | 4 2 | | | I I*a I*a | | | -------------- + -------------- - -------------- otherwise | | | ________ 3/2 3/2 | | | / 2 / 2\ / 2\ | | | / a 4 | a | 2 | a | | | | / 1 - -- x *|1 - --| x *|1 - --| | | | / 2 | 2| | 2| | | \ \/ x \ x / \ x / | | |/ \0
$$\begin{cases} - \int\limits_{a}^{0} \begin{cases} \frac{a^{4}}{x^{4} \left(\frac{a^{2}}{x^{2}} - 1\right)^{\frac{3}{2}}} - \frac{a^{2}}{x^{2} \left(\frac{a^{2}}{x^{2}} - 1\right)^{\frac{3}{2}}} - \frac{1}{\sqrt{\frac{a^{2}}{x^{2}} - 1}} & \text{for}\: a^{2} \left|{\frac{1}{x^{2}}}\right| > 1 \\\frac{i a^{4}}{x^{4} \left(- \frac{a^{2}}{x^{2}} + 1\right)^{\frac{3}{2}}} - \frac{i a^{2}}{x^{2} \left(- \frac{a^{2}}{x^{2}} + 1\right)^{\frac{3}{2}}} + \frac{i}{\sqrt{- \frac{a^{2}}{x^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{for}\: a < 0 \\\int\limits_{0}^{a} \begin{cases} \frac{a^{4}}{x^{4} \left(\frac{a^{2}}{x^{2}} - 1\right)^{\frac{3}{2}}} - \frac{a^{2}}{x^{2} \left(\frac{a^{2}}{x^{2}} - 1\right)^{\frac{3}{2}}} - \frac{1}{\sqrt{\frac{a^{2}}{x^{2}} - 1}} & \text{for}\: a^{2} \left|{\frac{1}{x^{2}}}\right| > 1 \\\frac{i a^{4}}{x^{4} \left(- \frac{a^{2}}{x^{2}} + 1\right)^{\frac{3}{2}}} - \frac{i a^{2}}{x^{2} \left(- \frac{a^{2}}{x^{2}} + 1\right)^{\frac{3}{2}}} + \frac{i}{\sqrt{- \frac{a^{2}}{x^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
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/ 0 | / | | | | / 4 2 | | | 1 a a 2 |1 | | | |- --------------- + --------------- - --------------- for a *|--| > 1 | | | _________ 3/2 3/2 | 2| | | | / 2 / 2\ / 2\ |x | | | | / a 4 | a | 2 | a | | | | / -1 + -- x *|-1 + --| x *|-1 + --| | | | / 2 | 2| | 2| | | | \/ x \ x / \ x / |- | < dx for a < 0 | | | 4 2 | | | I I*a I*a | | | -------------- + -------------- - -------------- otherwise | | | ________ 3/2 3/2 | | | / 2 / 2\ / 2\ | | | / a 4 | a | 2 | a | | | | / 1 - -- x *|1 - --| x *|1 - --| | | | / 2 | 2| | 2| | | \ \/ x \ x / \ x / | | | / | a < | a | / | | | | / 4 2 | | | 1 a a 2 |1 | | | |- --------------- + --------------- - --------------- for a *|--| > 1 | | | _________ 3/2 3/2 | 2| | | | / 2 / 2\ / 2\ |x | | | | / a 4 | a | 2 | a | | | | / -1 + -- x *|-1 + --| x *|-1 + --| | | | / 2 | 2| | 2| | | | \/ x \ x / \ x / | | < dx otherwise | | | 4 2 | | | I I*a I*a | | | -------------- + -------------- - -------------- otherwise | | | ________ 3/2 3/2 | | | / 2 / 2\ / 2\ | | | / a 4 | a | 2 | a | | | | / 1 - -- x *|1 - --| x *|1 - --| | | | / 2 | 2| | 2| | | \ \/ x \ x / \ x / | | |/ \0
$$\begin{cases} - \int\limits_{a}^{0} \begin{cases} \frac{a^{4}}{x^{4} \left(\frac{a^{2}}{x^{2}} - 1\right)^{\frac{3}{2}}} - \frac{a^{2}}{x^{2} \left(\frac{a^{2}}{x^{2}} - 1\right)^{\frac{3}{2}}} - \frac{1}{\sqrt{\frac{a^{2}}{x^{2}} - 1}} & \text{for}\: a^{2} \left|{\frac{1}{x^{2}}}\right| > 1 \\\frac{i a^{4}}{x^{4} \left(- \frac{a^{2}}{x^{2}} + 1\right)^{\frac{3}{2}}} - \frac{i a^{2}}{x^{2} \left(- \frac{a^{2}}{x^{2}} + 1\right)^{\frac{3}{2}}} + \frac{i}{\sqrt{- \frac{a^{2}}{x^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{for}\: a < 0 \\\int\limits_{0}^{a} \begin{cases} \frac{a^{4}}{x^{4} \left(\frac{a^{2}}{x^{2}} - 1\right)^{\frac{3}{2}}} - \frac{a^{2}}{x^{2} \left(\frac{a^{2}}{x^{2}} - 1\right)^{\frac{3}{2}}} - \frac{1}{\sqrt{\frac{a^{2}}{x^{2}} - 1}} & \text{for}\: a^{2} \left|{\frac{1}{x^{2}}}\right| > 1 \\\frac{i a^{4}}{x^{4} \left(- \frac{a^{2}}{x^{2}} + 1\right)^{\frac{3}{2}}} - \frac{i a^{2}}{x^{2} \left(- \frac{a^{2}}{x^{2}} + 1\right)^{\frac{3}{2}}} + \frac{i}{\sqrt{- \frac{a^{2}}{x^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
Piecewise((-Integral(Piecewise((-1/sqrt(-1 + a^2/x^2) + a^4/(x^4*(-1 + a^2/x^2)^(3/2)) - a^2/(x^2*(-1 + a^2/x^2)^(3/2)), a^2*Abs(x^(-2)) > 1), (i/sqrt(1 - a^2/x^2) + i*a^4/(x^4*(1 - a^2/x^2)^(3/2)) - i*a^2/(x^2*(1 - a^2/x^2)^(3/2)), True)), (x, a, 0)), a < 0), (Integral(Piecewise((-1/sqrt(-1 + a^2/x^2) + a^4/(x^4*(-1 + a^2/x^2)^(3/2)) - a^2/(x^2*(-1 + a^2/x^2)^(3/2)), a^2*Abs(x^(-2)) > 1), (i/sqrt(1 - a^2/x^2) + i*a^4/(x^4*(1 - a^2/x^2)^(3/2)) - i*a^2/(x^2*(1 - a^2/x^2)^(3/2)), True)), (x, 0, a)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.